Mapping/Mistakes/Image Elements not in Codomain
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Example of Mistake in Definition of Mapping
This example of an attempted definition of a mapping contains a mistake.
- $f: \N \to \N$ defined as: $\forall x \in \N: x \mapsto x - 7$
Explanation
The codomain of $f$ is:
- $\Cdm f = \set {0, 1, 2, \ldots}$
However, consider the subset $S \subset \Dom f$ of the domain of $f$ where $S = \set {0, 1, 2, 3, 4, 5, 6}$:
- $f \sqbrk S = \set {-7, -6, -5, -4, -3, -2, -1}$
and none of the elements of the image of $S$ are actually elements of $\Cdm f$.
That is:
- $\Img f \nsubseteq \Cdm f$
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Mappings: Exercise $1 \ \text {(i)}$